Diffusion coefficients from sorption data

The validity of eqn. (3) for determining the intracrystalline self-diffusion coefficients from uptake data has been shown for the sorption of benzene by... 

D. M. Ruthven and K. F. Loughlin (University of New Brunswick, Fredericton, N. B., Canada) The data of Kondis and Dranoff illustrate a number of important points relating to the problem of calculating diffusion coefficients from sorption curves. For the analysis of sorption curves, the crystal size distribution should be expressed on a weight (or... 

Studies of the diffusion of benzene in natural rubber represent some of the earliest detailed examinations of the interaction of an organic solvent with a polymer. Hayes and Park carried out measurements at low concentrations by the vapor sorption method (1), and at higher concentrations by determining the concentration distribution using an interferometric method (2). Complementary measurements by vapor transmission to determine the diffusion coefficient from time-lag data were carried out at low concentrations by Barrer and Fergusson (3). The main results of these studies have been summarized in Fujita s review (4) of organic vapor diffusion in polymers above the glass transition temperature. However, the problems with these measurements were not referenced. 

The diffusivity data were obtained by two different methods i.e. from the time-lag in the gas permeation and from the sorption kinetics. The results from both methods are plotted versus SWNT loading in Fig. 8.80. Even though the absolute values obtained from the two different methods are different, the trends observed are the same i.e., there is a significant increase in diffusion coefficient from 0 to 5 wt% loading while the diffusion coefficient either leveled off or decreased slightly by the further increase in SWNT loading. 

An elegant alternative method to measure sorption into polymers is the Attenuated Total Reflectance Fourier Transform Infrared (ATR-FTIR) method. It allows in situ acquisition of the kinetic data and at the same time records the changes that occur in the polymer matrix due to the influence of the diffusant. Effects such as swelling, changes in morphology and polymer solvent interactions can all be simultaneously monitored. To calculate the diffusion coefficients from ATR-FTIR data, the mass uptake equation used in gravimetric diffusion experiments has to be modified to take into account the convolution of the evanescent field with the diffusion profile. 

Pure PHEMA gel is sufficiently physically cross-linked by entanglements that it swells in water without dissolving, even without covalent cross-links. Its water sorption kinetics are Fickian over a broad temperature range. As the temperature increases, the diffusion coefficient of the sorption process rises from a value of 3.2 X 10 8 cm2/s at 4°C to 5.6 x 10 7 cm2/s at 88°C according to an Arrhenius rate law with an activation energy of 6.1 kcal/mol. At 5°C, the sample becomes completely rubbery at 60% of the equilibrium solvent uptake (q = 1.67). This transition drops steadily as Tg is approached ( 90°C), so that at 88°C the sample becomes entirely rubbery with less than 30% of the equilibrium uptake (q = 1.51) (data cited here are from Ref. 138). 

Figure 15 The sorption of acetaminophen from a solution of limited volume by 10 X 4 poly (A-isopropylacry lam idc) gel, illustrating the use of Eqs. (34) and (35) to determine the diffusion coefficient of the solute. (Data from Ref. 174.)... 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

In Section IB we presented experimental evidence that diffusion coefficients correlate with PVC main-chain polymer motions. This relationship has also been justified theoretically (12). In the previous section we demonstrated that the presence of CO2 effects the cooperative main-chain motions of the polymer. The increase in with increasing gas concentration means that the real diffusion coefficient [D in eq. (11)] must also increase with concentration. The nmr results reflect the real diffusion coefficients, since the gas concentration is uniform throughout the polymer sample under the static gas pressures and equilibrium conditions of the nmr measurements. Unfortunately, the real diffusion coefficient, the diffusion coefficient in the absence of a concentration gradient, cannot be determined from classical sorption and transport data without the aid of a transport model. Without prejustice to any particular model, we can only use the relative change in the real diffusion coefficient to indicate the relative change in the apparent diffusion coefficient. 

In the derivation of the simplified expressions for solubility and diffusion coefficients, eqs. (4) and (9), C was assumed to be small. This fact does not limit the usefulness of these expressions for high concentrations. We show below that sorption and transport expressions, eqs. (11) and (14), respectively, derived from the simplified equations retain the proper functional form for describing experimental data without being needlessly cumbersome. Of course, the values of the parameters in eqs. (4) and (9) will differ from the corresponding parameters in eqs. (3) and (8), to compensate for the fact that the truncated power series used in eqs. (4) and (9) poorly represent the exponentials when aC>l or 0C>1. Nevertheless, this does not hinder the use of the simplified equations for making correlation between gas-polymer systems. 

Wonders and Paul (15) report that a nonlinear least-squares fit of the dual-mode expression [eq. (16) in the preceding chapter] to the permeability versus pressure data, for C02 in polycarbonate, gives and values of 4.78 x 10 8 and 7.11 x 10 9 cm2/sec, respectively. The broken curve in Fig. 2 was calculated from the dual-mode sorption coefficients of Fig. 1 and the values of the diffusion coefficients given above. 

As a general rule, membrane material changes affect the diffusion coefficient of a permeant much more than the sorption coefficient. For example, Figure 2.18 shows some typical gas permeation data taken from a paper of Tanaka et al. [23], The diffusion and sorption coefficients of four gases in a family of 18 related polyimides are plotted against each other. Both sorption and diffusion coefficients... 

Figure 2.25 Diffusion and sorption coefficients of methane in different families of polymer materials. Diffusion coefficients change over a wide range but sorption coefficients are relatively constant. Data from references [23,35-37]...

Lewin and coworkers [255-260] developed an accessibility system based on equilibrium sorption of bromine, from its water solution at pH below 2 and at room temperature, on the glycosidic oxygens of the cellulose. The size of the bromine molecule, its simple structure, hydrophobicity, nonswelling, and very slow reactivity with cellulose in acidic solutions, contribute to the accuracy and reproducibility of the data obtained. The cellulose (10 g/1) is suspended in aqueous bromine solutions of 0.01-0.02 mol/1 for 1-3 h, depending on the nature of the cellulose, to reach sorption equilibrium. The diffusion coefficients of bromine in cotton and rayon are 4.6 and 0.37 x 10 cm /min, respectively. The sorption was found to strictly obey the Langmuir isotherm, which enables the calculation of the accessibility of the cellulose as follows ... 

Fig. 18. Self-diffusion coefficients of benzene in NaX at 458 K PFG NMR, O (97) and (92) (JENS, A (13) deduced from NMR lineshape analysis, (10). Comparison with nonequilibrium measurements T, sorption uptake with piezometric control (93) , zero-length column method (96) o, frequency-response and single-step frequency-response technique (98). The region of the results of gravimetric measurements with different specimens (92) is indicated by the hatched areas. Asterisked symbols represent data obtained by extrapolation from lower temperatures with an activation energy confirmed by NMR measurements.

Membrane phase concentration of component i in the feed side, Cg, can be calculated from its bulk concentration by Henry s equation (Equation 5.8) provided it is present in trace amount in the feed solution. For higher concentration of component i, Cg can be obtained from experimental sorption data. Membrane phase concentration on the permeate side of component i, i.e., Cpi may be neglected due to the low pressure the activity of the component in the downstream side is very low. Thus, Equation 5.28 can be readily solved to calculate the theoretical flux and diffusion coefficient of i or j component employing any of the above equations relating the diffusion coefficient and concentration. Equations 5.14 through 5.25 depending on its best matching with the experimental data. 

The sorption/desorption experiments were carried out as a function of a both above and below the Tg of the matrix. The data obtained from both experiments closely resembled Fickian diffusion. Below 0.55a (between 0 < Mt and Mf < 0.5) and above 0.55a (between 0.08 < Mf and Mf < 0.75) initial slope of Mf/Mf vs. curves were linear with respect to abscissa (R > 0.98). Diffusion coefficients were obtained using the linear portion of the normalized moisture sorption (Mt/Moo vs. curves from Equation 46.14. 

Comparison between experimental diffusion data from the sorption (circles) and desorption (triangles) and predicted (solid lines) diffusion coefficients for water. 

It is possible that the lower than required values of D2 reflect a problem with incorrect values of Q, which if too large would result in smaller values of D2. In an interferometric study of the diffusion of toluene in an uncrosslinked natural rubber sample, Mozisek (15) reported results for the mutual diffusion coefficient which were similar to the results of Hayes and Park. In the absence of thermodynamic data from Mozisek s work, correction factors calculated for the present work were applied to his data. The results are shown in Figure 7, which reproduces Mozisek s data along with the values for D2. The extrapolated value at 1, would exceed the self diffusion coefficient for toluene by about two orders of magnitude, similar to the discrepancy seen with Hayes and Park s data. This indicates that the fault with the results in the present case is not due to overly high values of the correction factors. Moreover, the method of calculating D from D12 has been confirmed experimentally by Duda and Vrentas (16) in a comparison of vapor sorption results for toluene diffusion in molten polystyrene with the values of D1 obtained directly using radio-labeled toluene. 

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